Arc Measure: Definition & Formula

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  • 0:04 Arc of a Circle
  • 0:37 Arc Measure vs Arc Length
  • 1:10 Degrees and Radians
  • 2:46 Arc Measure Formula
  • 3:46 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Arc measure is a way of identifying an arc. In this lesson, we'll learn how to define and calculate arc measure while discussing the different elements involved in this calculation.

Arc of a Circle

Let's start with a simple definition. An arc is a segment of a circle. For example, you've probably seen a rainbow at some point in your life, so imagine using a giant piece of paper to draw a full circle that includes the rainbow. Do you see how rainbow is a segment of a circle?

Rainbows Are Arcs
Arc measure 1

As we can see in the picture above, rainbows come in all different lengths, as do arcs. An arc can be any length up to a full circle; therefore, it's important for us to know how to identify and measure different arcs. Let's take a few minutes to learn how to do that now.

Arc Measure vs. Arc Length

There are two different ways to identify an arc. We can identify it by its length or by the measure of the angle that the arc creates in the center of a circle. The span of the arc is called the arc length, and the measure of the angle that the arc creates is called the arc measure. Check out the picture below, which illustrates both characteristics of an arc:

Arc of a Circle
Circle 1

To understand an arc measure, we need to be familiar with the measure of an angle in both degrees and radians. Let's now talk for a bit about degrees and radians and the relationship between the two.

Degrees and Radians

Angles have two different units of measure. A circle measures 360 degrees. The degree of an angle will represent the same fraction of a circle as the angle's corresponding arc. For example, 90 degrees is 1/4 of 360 degrees, so a 90-degree angle has a corresponding arc that is 1/4 of a circle.

The other unit of measurement we use to measure an angle is the radian. The relationship between radians and degrees allows us to convert between the two using the rules below. Keep in mind that a full circle equals 360 degrees, or 2pi radians, and that a single radian equals 180 degrees.

  • The measure of an angle in degrees equals the measure of an angle in radians: To convert degrees to radians, we multiply the degree measure by pi / 180.
  • The measure of an angle in radians equals the measure of an angle in degrees: To convert radians to degrees, we multiply the radian measure by 180 / pi.

For example, let's calculate the radian measure of a 90-degree angle.

  • 90 multiplied by pi / 180 = 90(pi / 180).
  • 90pi / 180 = pi / 2 radians.

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